Sufficient conditions

Naturally, we would like to determine the optimum that is global, i. e., valid over the entire space of admissible control functions. This quest urges us to inquire into the conditions that are sufficient for the global optimum, i. e., the optimum over the entire space of admissible control functions. 

For the minimum solution to be global, a sufficient condition must ensure that the objective functional value is the lowest possible. In the context of an optimal control problem, a sufficient condition for the minimum should lead to the inequality 

To yield the above inequality, a sufficient condition must include at least one inequality, which in fact is another optimal control problem. Thus, for the global minimum in the simplest optimal control problem, a set of sufficient conditions by Mangasarian (1966) requires that 

the functions F [Equation (3.31), p. 67] and g are convex jointly with respect to the states and corresponding controls 

The third requirement of Mangasarian gives rise to inequalities of the form E(y, u) 0 [see Inequality (3.52), p.82]. Proving it becomes a new optimal control problem in which zero needs to be estabhshed as the minimum of L for all admissible u. This task cannot be accomplished analytically except in rare, simple cases where the Hessian of L could be easily checked for positive definiteness. Because of this situation, the application of any sufficient condition in optimal control is very limited. 

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Smith B.D. Image reconstruction from cone-beam projections necessary and sufficient conditions and reconstruction methods., IEEE Trans. Med. Imaging, V. 4, 1985, p. 14-28. 

Using the theorem that the sufficiency condition for mathematical correctness in 3D-reconstruction is fulfilled if all planes intersecting the object have to intersect the source-trajectory at least in one point [8], it is possible to generalise Feldkamp s method. Using projection data measured after changing the sotuce-trajectory from circular to spiral focus orbit it is possible to reconstruct the sample volume in a better way with the Wang algorithm [9]. 

Another efficient and practical method for exact 3D-reconstruction is the Grangeat algorithm [11]. First the derivative of the three-dimensional Radon transfomi is computed from the Cone-Beam projections. Afterwards the 3D-Object is reconstructed from the derivative of the Radon transform. At present time this method is not available for spiral orbits, instead two perpendicular circular trajectories are suitable to meet the above sufficiency condition. 

Appendix A The Jahn-Teller Model and the Longuet-Higgins Phase Appendix B The Sufficient Conditions for Having an Analytic Adiabatic-to-Diabatic Transformation Matrix I. Orthogonality II. Analyticity... 

Notice that the necessaiy and sufficient condition for this matrix to become diagonal is that the following condition ... 

In Section in.B, and later in Appendix C, it was shown that the sufficient condition for the adiabatic-to-diabatic transfonnation matrix A to be single valued in a given region in configuration space is the fulfillment of the following curl condition [8,34] ... 

APPENDIX B THE SUFFICIENT CONDITIONS FOR HAVING AN ANALYTIC ADIABATIC-TO-DIABATIC TRANSFORMATION MATRIX... 

To pursue this question we shall examine the stability of certain steady state solutions of Che above equaclons by the well known technique of linearized stability analysis, which gives a necessary (but noc sufficient) condition for the stability of Che steady state. 

The written directives of a quality control program are a necessary, but not a sufficient, condition for obtaining and maintaining an analysis in a state of statistical control. Although quality control directives explain how an analysis should be properly conducted, they do not indicate whether the system is under statistical control. This is the role of quality assessment, which is the second component of a quality assurance program. 

Now let us prove two theorems containing sufficient conditions of weak lower semicontinuity of the functionals. 

Mechanical properties of mbber-modifted epoxy resins depend on the extent of mbber-phase separation and on the morphological features of the mbber phase. Dissolved mbber causes plastic deformation and necking at low strains, but does not result in impact toughening. The presence of mbber particles is a necessary but not sufficient condition for achieving impact resistance. Optimum properties are obtained with materials comprising both dissolved and phase-separated mbber (305). 

CUO2 layers appear in all cuprate superconductors and appear to be a necessary but not sufficient condition for high temperature superconduction. The La2SrCu20g 2 compound has CUO2 layers but does not superconduct. Experiments also indicate that T is proportional to the carrier density in the CUO2 layer but not to the volume carrier density, which is further evidence that the YBa2Cu202 is a two-dimensional superconductor. 

Theorem 5. The transpose of is a complete B-matrrx of equation 13. It is advantageous if the dependent variables or the variables that can be regulated each occur in only one dimensionless product, so that a functional relationship among these dimensionless products may be most easily determined (8). For example, if a velocity is easily varied experimentally, then the velocity should occur in only one of the independent dimensionless variables (products). In other words, it is sometimes desirable to have certain specified variables, each of which occurs in one and only one of the B-vectors. The following theorem gives a necessary and sufficient condition for the existence of such a complete B-matrix. This result can be used to enumerate such a B-matrix without the necessity of exhausting all possibilities by linear combinations. 

Sufficient Conditions for the Existence of Laplace Transform Suppose/ is a function which is (1) piecewise continuous on eveiy finite intei val 0 < t exponential growth at infinity, and (3) Jo l/t)l dt exist (finite) for every finite 6 > 0. Then the Laplace transform of/exists for all complex numbers. s with sufficiently large real part. 

Sufficient conditions are that any local move away from the optimal point ti gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point ti up to second-order terms ... 

If ti satisfies necessaiy conditions [Eq. (3-80)], the second term disappears in this last line. Sufficient conditions for the point to be a local minimum are that the matrix of second partial derivatives F is positive definite. This matrix is symmetric, so all of its eigenvalues are real to be positive definite, they must all be greater than zero. 

Sufficiency conditions to assure that a Kuhn-Tucker point is a local minimum point require one to prove that the objec tive function will increase for any feasible move away from such a point. To carry out such a test, one has to generate the matrix of second derivatives of the Lagrange func tion with respect to all the variables z evaluated at z. The test is seldom done, as it requires too much work. 

A sufficient condition for a unique Newton direction is that the matrix of constraint derivatives is of full rank (linear independence... 

If (a) above is satisfied, then the necessary and sufficient condition for stability is either... 

The important beneficial effects that substrate roughness can bring were firmly established in the late sixties and early seventies, principally as a result of work in two areas. The first was associated with the electroless deposition of metals onto plastics such as ABS and polypropylene. In the process the plastics must be etched in a way which produces pits on a micrometre scale. Such a topography had been shown to be a necessary, but not sufficient condition for adequate adhesion [40]. 

It is an extremely difficult task to establish generally valid sufficient conditions for roughness-induced wetting. This is a direct consequnce of the loss of translational invariance in such systems. A vast majority of the hitherto performed calculations [185,201] have been based on a simplified model of a rough substrate which assumes periodic variation of the substrate surface location. 

The basic requirement is that the time variation of Eq. (10) should vanish in the thermodynamic equilibrium. A sufficient condition for this is the principle of the detailed balance, which means that each term in the sum on the right-hand side of Eq. (10) vanishes for an equilibrium probability density t)eq... 

Hammett originally demonstrated that the sigma-rho relation might be expected to hold if entropies of activation (or entropy changes) were constant in a series. It has since been shown that a sufficient condition is a linear relation between enthalpies and entropies of activation (or reaction), and such hnear relations are frequently encountered. Although the existence of such linear relations has always appeared somewhat mysterious, some rationale for this relationship has recently been given. 

The above theories are a necessary condition to judge the compatibility between two polymers, but not a sufficient condition. 

The following theorem - stated without proof (see [jen86a]) - gives necessary and sufficient conditions for CA rules to generate constant temporal sequences ... 

A sufficient condition for fo to solve equation 9.33 is that the term in the curly brackets itself vanishes ... 

It should be emphasized that not all normalizable hermitean matrices r(x x 2. . . x xlx2. . . xp) having the correct antisymmetry property are necessarily strict density matrices, i.e., are derivable from a wave function W. For instance, for p — N, it is a necessary and sufficient condition that the matrix JT is idempotent, so that r2 = r, Tr (JH) = 1. This means that the F-space goes conceptually outside the -space, which it fully contains. The relation IV. 5 has apparently a meaning within the entire jT-space, independent of whether T is connected with a wave function or not. The question is only which restrictions one has to impose on r in order to secure the validity of the inequality... 

For simple salts the influence of parameters (1)—(3) can be studied separately by the investigation of series of salts with a common anion or cation in a solvent of high dielectric permittivity. Flowever, high solvent permittivity is only a necessary, but not a sufficient, condition for complete dissociation. High permittivity of the solvents does not prevent ions from associating, if these ions interact specifically... 

Let D be a nonsingular diagonal matrix, and apply the theorem proved above to the matrix D — B. This matrix is nonsingular if and only if I — D 1B is nonsingular, and a sufficient condition for this is... 

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